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arXiv:2411.11041 · Journal of Applied and Numerical Analysis

Directional diffusion splitting method for advection-diffusion-reaction model

R. H. Drebotiy, H. А. Shynkarenko

Published Sep 16, 2025• 0 citations
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Abstract

In the context of investigating the potential application of quantum computers to the finite element method, we propose a specific approach for solving the two-dimensional parabolic advection-diffusion-reaction boundary value problem by transforming it into a set of corresponding one-dimensional problems. This method employs specialized operator splitting, resulting in a recurrent fractional-step scheme for time integration. While this scheme offers significant parallelization potential, the main goal is to map the original large elliptic 2D problem arising at each time step into a set of smaller 1D problems of the same kind. This transformation, combined with recently proposed quantum-assisted stabilization schemes, allows us to pass smaller linear systems to a quantum linear solver, making them suitable for near-term quantum computers. We combine the proposed algorithm with the finite element method to solve the resulting one-dimensional problems, although these problems can also be addressed using other discretization techniques, such as the finite volume or finite difference methods.

DOI
10.30970/ana.2025.3.35

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In the context of investigating the potential application of quantum computers to the finite element method, we propose a specific approach for solving the two-dimensional parabolic advection-diffusion-reaction boundary value problem by transforming it into a set of corresponding one-dimensional problems. This method employs specialized operator splitting, resulting in a recurrent fractional-step scheme for time integration. While this scheme offers significant parallelization potential, the main goal is to map the original large elliptic 2D problem arising at each time step into a set of smaller 1D problems of the same kind. This transformation, combined with recently proposed quantum-assisted stabilization schemes, allows us to pass smaller linear systems to a quantum linear solver, making them suitable for near-term quantum computers. We combine the proposed algorithm with the finite element method to solve the resulting one-dimensional problems, although these problems can also be addressed using other discretization techniques, such as the finite volume or finite difference methods.
Quantum computing is a rapidly advancing field, driven by the potential advantages derived from the unique properties of quantum entanglement. In particular, the exponential speedup of certain carefully designed algorithms, compared to their classical counterparts, promises to significantly enhance the numerical solution of a wide range of problems. This paper investigates the integration of quantum computing with the finite element method, focusing on singularly perturbed advection-diffusion-reaction problems. We introduce a novel finite element scheme that combines classical and quantum algorithms. In this approach, the primary mesh adaptation loop is managed by a classical computer, while a specific stabilization procedure is executed on a quantum computer. This procedure leverages the Harrow-Hassidim-Lloyd algorithm in conjunction with the swap test to estimate the value of a linear functional, which constitutes a substantial portion of the computational workload. We demonstrate that this hybrid approach effectively eliminates parasitic oscillations in the finite element approximation, even at the early stages of the adaptation process. This leads to a significant improvement in the quality of intermediate finite element solutions. As a result, our scheme offers more efficient feedback with reduced computational costs for researchers using the method to investigate physical phenomena. To support the scheme, we prove special explicit a posteriori error estimates. Possible benefits of the proposed finite element scheme are analyzed using the numerical comparison with the typical adaptive scheme.
We present a versatile and efficient quantum algorithm based on the Lattice Boltzmann method (LBM) approximate solution of the linear advection-diffusion equation (ADE). We emphasize that the LBM approximation modifies the diffusion term of the underlying exact ADE and leads to a modified equation (mADE). Due to its versatility in terms of operator splitting, the proposed quantum LBM algorithm for the mADE provides a building block for future quantum algorithms to solve the linearized Navier-Stokes equation on quantum computers. We split the algorithm into four operations: initialization, collision, streaming, and calculation of the macroscopic quantities. We propose general quantum building blocks for each operator, which adapt intrinsically from the general three-dimensional case to smaller dimensions and apply to arbitrary lattice-velocity sets. Based on (sub-linear) amplitude data encoding, we propose improved initialization and collision operations with reduced complexity and efficient sampling-based simulation. Quantum streaming algorithms are based on previous developments. The proposed quantum algorithm allows for the computation of successive time steps, requiring full state measurement and reinitialization after every time step. It is validated by comparison with a digital implementation and based on analytical solutions in one and two dimensions. Furthermore, we demonstrate the versatility of the quantum algorithm for two cases with non-uniform advection velocities in two and three dimensions. Various velocity sets are considered to further highlight the flexibility of the algorithm. We benchmark our optimized quantum algorithm against previous methods employed in sampling-based quantum simulators. We demonstrate sampling efficiency, with sampling accelerated convergence requiring fewer shots. • Novel LBM quantum algorithm for solving the linear advection-diffusion equation with variable velocity as a building block. • Modular quantum algorithm for 1 through 3 spatial dimensions and all common lattice-velocity sets. • Significant reduction in quantum-algorithm complexity achieved through efficient encoding strategy leveraging symmetries. • Improved collision quantum-algorithm building block leading to higher success rates for the collision operator. • Benchmarking shows faster convergence and reduced shot requirements compared to previous algorithms.
Simulating differential equations on classical computers becomes an intractable problem if the grid size is extremely large. Quantum computers are believed to achieve a possibly exponential speedup in the matrix operation. In this paper, we propose a quantum algorithm for solving the advection-diffusion-reaction equation by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. First, the effectiveness of the proposed approximate PITE operator is justified by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit to realize the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. Compared to the existing algorithms for the quantum linear system problem, our algorithm achieves an exponential speedup regarding the matrix size at the cost of a worse dependence on the error bound. Moreover, numerical simulations using gate-based quantum emulator for 1D/2D examples are also provided to verify our algorithm. Finally, we extend our algorithm to the coupled system of advection-diffusion equations to show the prospects for practical applications.
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the spatial dimension, a quantum computer can produce a quantum state that encodes the solution with only polynomial complexity, provided that suitable input access is available. In this work, we investigate efficient quantum algorithms for linear and nonlinear fractional reaction-diffusion equations with periodic boundary conditions. For linear equations, we analyze and compare the complexity of various methods, including the second-order Trotter formula, time-marching method, and truncated Dyson series method. We also present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism, resulting in optimal scaling in the spatial dimension. For nonlinear equations, we employ the Carleman linearization method and propose a block-encoding version that is appropriate for the dense matrices that arise from the spatial discretization of fractional reaction-diffusion equations.
Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.